Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 59


$$\cos74^\circ=\cos60^\circ\cos14^\circ+\sin60^\circ\sin14^\circ$$ The statement is false.

Work Step by Step

$$\cos74^\circ=\cos60^\circ\cos14^\circ+\sin60^\circ\sin14^\circ$$ As $74^\circ=60^\circ+14^\circ$, $$\cos74^\circ=\cos(60^\circ+14^\circ)$$ Now we use the cosine sum identity: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ (be extremely careful about the sign in the middle) That means $$\cos74^\circ=\cos60^\circ\cos14^\circ-\sin60^\circ\sin14^\circ$$ As you see, the sign in the middle is different: $$\cos60^\circ\cos14^\circ-\sin60^\circ\sin14^\circ\ne\cos60^\circ\cos14^\circ+\sin60^\circ\sin14^\circ$$ Therefore, the statement $$\cos74^\circ=\cos60^\circ\cos14^\circ+\sin60^\circ\sin14^\circ$$ is false.
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